_{1}

Chambers et al . (2014) set forth a decomposition of the Lerner index, which results in a function on the full space o f input and output prices and quantities, such that the effect of the Farrell output measure of technical efficiency is explicit. In close correspondence, a decomposition of the Lerner index is established in which allocative efficiency (in both standard and reversed form, as defined by Bogetoft et al., 2006) complements the effect of input technical efficiency, with the reversed decomposition bound to the hypothesis of homotheticity. The resulting functions on are conjectured to define pregnant perspectives on the benchmark relevance of homothetic models, and their generalizations to multiple output.

The scale symmetry in input (consumption) space embodied by homothetic production (utility) functions has long been recognized as a benchmark setting for the microeconomics of the producer (consumer). According to Chambers and Mitchell (2001) [

In turn, input homothetic correspondences represent benchmark models in the input radial perspective on productive efficiency, for which Farrell (1957) [

The plan of the rest of the paper is as follows. In Section 2 we recall the decomposition set forth by Chambers et al. (2014) [

In a seminal paper, Lerner (1934) [

being p the price of the single output y, w the price vector for the input vector x, and C and R the cost and revenue functions respectively. Thus, the above measures of economic performance provide a sound factorization of fundamental determinants of market power. Noticeably, a key to such a decomposition is given by the identity

takes a value lower than 1, the input bundle x represents an inefficient production plan for the output level y, the lower D_{o} the larger the inefficiency. Then, formula (1) fixes the effect of technical inefficiency on the Lerner index, as modulated by _{o} and R dual quantities). The index (1) is a function on the product of the (primal) technology set and the (dual) space of input and output prices; write X for such a space, domain of L.

Noticeably, the Authors establish decomposition (1) in the context of an approach to the Lerner index as the first order derivative of the Nerlovian indicator, upon restricting to single output and fixing the radial perspective in input space (g_{x} = 0 in the Nerlovian indicator). That being the case, homothetic production functions embody the well known properties resulting from input scale symmetry (straight expansion paths and separability of the cost function) which, in turn, fix a benchmark setting for the measurement of overall productive efficiency, as discussed in the following section.

In a celebrated paper, Farrell (1957) [

In a recent article, Bogetoft et al. (2006) [_{i} the input distance function), and then establishing allocative efficiency AE in ratio form as

([

The authors define a “reverse Farrell approach”, in which allocative efficiency is established first, and technical efficiency is subsequently fixed by projecting the allocative efficient input bundle onto the efficient frontier. The authors themselves discuss several cases in which organizations may find it optimal to pursue allocative efficiency irrespective of, or at least prior to, technical efficiency; for instance, “it may be easier to reallocate resources within a hierarchy or via markets, than to actually change the production procedures” (ivi, p. 451).

Thus, Bogetoft et al. (2006) [

and reversed technical efficiency subsequently, and then prove that standard and reversed Farrell decompositions do coincide for (input and input ray-) homothetic technologies. Correspondingly, exploiting the well known isomorphism between the microeconomic problems of the producer and of the consumer, Mantovi (2013) [

How about the consistency of employing the above measures of allocative efficiency in order to reshape the decomposition (1)? True, the technical efficiency measure employed by Chambers et al. (2014) [^{*} in [

As expected, no problem arises for standard Farrell decompositions: via definition (3) the decomposition (1) can be written

in which we factorize the effects of the overall efficiency TE×AE and of the expenditure/revenue ratio wx/py.

Consider then the reversed measures AE^{*}, TE^{*}: following Bogetoft et al. (2006) [

sides of the Chambers-Färe-Grosskopf decomposition (1) by

with^{*} of technical efficiency if and only if the production function is homothetic (as is well known, that being the case, the cost function can be written as the product of a function of w and a function of y, and therefore the ratio

To sum up, overall productive efficiency seems to tailor pregnant parallels to the Chambers-Färe-Grosskopf decomposition of the Lerner index as a function L on X. On conceptual grounds, such a function establishes a fundamental link between production analysis and industrial economics; on analytical grounds, the full range of variables (both primal and dual) which span X enables us in principle to employ the full theory of duality in order to express L in terms of the relevant quantities. In such respects, our expressions (5) and (6) of L establish a connection with the celebrated Farrell decomposition of overall productive efficiency, and in fact with the interesting advances set forth by Bogetoft et al. (2006) [

Still, the single output case may prove quite interesting for deepening the properties of nonhomothetic models. In such respects, Mantovi (2013) [

emerges in the definition

[